Research Pages Robert F. Mudde
Computed TomographyComputed Tomography is the technique by which a slice through the object of study is made visible via multiple Xray viewing angles. There are many algorithms for reconstructing an image from the different measured views. Some are fast but in accurate, other methods are slow but generate much sharper images. There are direct methods and iterative ones. The choice depends on what the aim of the tomographic image is and on the number of independent measurements compared to the number of pixels of the image. Generally, the higher the ratio of measured data over images pixels, the better the reconstructed image. 
List of algorithms
Direct MethodsGenetics 
LambertBeer LawThe basic idea of CT is to measure over a large number of different lines or 'tubes' the transmission of Xrays. The transmission
for monochromatic Xrays is governed by the LambertBeer law. It describes the Xray intensity when a Xray beam with intensity
I_{0} travels through a homogeneous medium of thickness x: 

Reconstruction ProblemAll measured raysums needs to be coupled to the powder distribution in the imaging plain. After the transformation from
measured intensity to amount of powder, a direct connection between the fraction of powder in the pixels traveled through by the xray
and the raysum can be made: The reconstruction thus boils down to solving the above equation, which can be written in vector/matrix form. 
Linear Back ProjectionA simple and fast algorithm to find an approximate solution to the above equation is the socalled linear back projection. Here, the inverse of W is estimated by taking its transpose W^{T}. A disadvantage of this approach is that it excessively blurs the reconstructions. An example of the performance of LBP is given in the figures to the right. The first figure shows the setup: the small circle in the middle is the fluid bed with three Xray sources around it. The red lines indicate the measuring rays from source to detectors. The next figure shows the original 'bubbles' cut by the measuring plane: one big bubble and one small one. Also, the discretized version of the bubbles is shown: air is black, powder is white. From the discretize version, the raysums (i.e. measured signals) are calculated. These data are used in the LBPreconstruction. A 45*45 grid has been used for both the discretization and the reconstruction. The result is seen in the next figure, showing that LBP produces poor images. 
Algebraic Reconstruction TechniqueAlternatively, images can be reconstructed using the Algebraic reconstruction Technique (ART). It uses an iterative scheme to find the best solution. In essence it tries to find the point in the multidimensional reconstruction space, that is closed to all hyperplanes defined by p = W . α. It is easily explained for a 2by2 grid. Then the above equation can be written as 
Genetic AlgorithmA completely different approach to reconstruction of the images is taken by the socalled Genetic Algorithms. The idea behind the genetic algorithms is to mimic nature and its evolution. Rather than solving a set of mathematical equations that describe the relation between the measured data and the object, it works with a large set of potential solutions that are seen as parents that have children. These children are genetic combinations of their parents with a survival of the fittest that steers the subsequent generation toward the best reconstructed image. The image is represented by a long vector of pixel values. In the genetic algorithm such a vector is seen as a chromosome, with the pixel values representing the genes. Every new generation is formed by crossing the genes of two parents. Parents are selected based on how well they already fit the measured data: the better the fit the more likely that such a gene will be chosen as parent. To avoid trapping in a local minimum, the children undergo mutation. Strange as it may sound, this procedure will eventually arive at a good image reconstruction. A simple example of a genetic algorithm in action can be seen on this page. 